<-y'/>- : -\ iJ:.'*,-- 



J : A J v - lO 



WESTLUND 












- 



LIBRARY OF CONGRESS. 

Ghapi*8£ Copyright No 

ShelL.Jvi.4- 

UNITED STATES OF .AMERICA. 



Digitized by the Internet Archive 
in 2011 with funding from 
The Library of Congress 



http://www.archive.org/details/outlinesoflogicOOwest 



X 



Outlines of Logic. 



BY 



Jacob Westlund, 

PROFESSOR OF MATHEMATICS IN BETHANY COLLEGE, 

LINDSBORG, KANSAS. 



1 MAY 10 1 896 ) 



TOPEKA, KANSAS: 

CRANE & CO., PUBLISHERS. 

18 9 6. 



^ 



4- 



[THE LltlAl 
[qrC ONGfr SMl 

WA«■H^M?I£!f ,) ' , 



Copyright, 1896, by Crane & Co. 



TABLE OF CONTENTS. 



INTRODUCTION. 

DEFINITION AND SCOPE OF LOGIC. 

PAGE. 

1. Definition of Logic 1 

2. Utility of Logic .'. . 2 

3. Operations of the Mind .' 2 

CHAPTER I. 

fundamental laws of thought. 

1. Definition of Law of Thought 4 

2. Fundamental Laws of Thought 4 

1. Law of Identity 4 

2. Law of Contradiction 5 

3. Law of Excluded Middle 5 

4. Law of Sufficient Reason 5 

CHAPTER II. 

concepts. 

1. Definition of Concept 7 

2. Classification of Concepts 7 

3. Content and Extent 8 

4. Relation of Concepts : 

1. Compatible Concepts 10 

2. Incompatible Concepts 10 

3. Subordinate Concepts 11 

4. Coordinate Concepts 12 

(iii) 



JV OUTLINES OF LOGIC. 

CHAPTER III. 

JUDGMENTS. 

PAGE. 

1. Definition of Judgment 14 

2. Terms 14 

3. Classification of Judgments: 

1. Quality 15 

2. Quantity 15 

3. Relation 17 

4. Modality 19 

4. Distribution of Teijms 20 

5. Immediate Inference : 

1. Synonymous Judgments 23 

2. Subalterns 24 

3. Opposition 25 

4. Conversion 29 

6. Simple and Complex Propositions 33 

CHAPTER IV. 

syllogisms. 

1. Definition of Syllogism 35 

2. Classification of Syllogisms 36 

3. Categorical Syllogisms : 

1. Definition 37 

2. Rules 37 

3. Explanation of Rules 38 

4. Figures , 42 

5. Moods 43 

4. Hypothetical Syllogisms: 

1. Definition 49 

2. Moods 50 

3. Fallacies 52 

4. Reduction of Hypothetical to Categorical Syllogisms 53 

5. Disjunctive Syllogisms: 

1. Definition 54 

2. Rules 54 

3. Moods 54 



TABLE OF CONTENTS. 



PAGE. 

6. DILEMMA 55 

7. Compound Syllogisms 56 

8. Abridged Syllogisms : 

1. Enthymeme 58 

2. Epichirema 59 

3. Sorites 59 

CHAPTER V. 
fallacies. 

1. Definition of Fallacy 63 

2. Classification of Fallacies 63 

3. Logical Fallacies : 

1. Fallacy of Equivocation 65 

2. Fallacy of Composition 66 

3. Fallacy of Division 66 

4. Fallacy of Accident 67 

5. Converse Fallacy of Accident <>7 

6. Fallacy of Many Questions 67 

7. Fallacy of Amphibology 68 

8. Fallacy of Positive and Negative Intention 68 

4. Material Fallacies : 

1. Begging the Question 69 

2. Fallacy of False Cause 69 

3. Fallacy of Irrelevant Conclusion 70 

5. Paralogisms and Sophisms 70 

CHAPTER VI. 

METHOD. 

1. Science : 

1. Definition of Science 72 

2. Requisites of a Science 72 

3. Axioms 73 

2. Deduction and Induction : 

1. Definition of Method 73 

2. Deduction 74 

3. Induction 74 



vi OUTLINES OF LOGIC. 

3. Definition : page. 

1. Definition defined 76 

2. Rules for Definition 77 

3. Nominal and Real Definitions 80 

4. Description 80 

4. Division : 

1. Division defined 81 

2. Dichotomy 81 

3. Rules lor Division 82 

4. Partition 84 

5. Demonstration: 

1. Demonstration defined 85 

2. Rules for Demonstration 85 

3. Classification of Demonstrations 86 

6. Analogy 90 

7. Hypothesis 91 

8. Classification of Sciences 91 



EXERCISES 92-102 



INTRODUCTION. 



DEFINITION AND SCOPE OF LOGIC. 
1. Definition of Logic. 

Logic is the science of the formal laios of human 
thought. 

Logic is the science which has for its object to inves- 
tigate the laws of human thought apart from the other acts 
of the mind. It explains the laws and principles by which 
all reasoning must be governed. In all sciences the rea- 
soning must be in accordance with the principles of logic, 
and although the method may be different in different 
sciences it must always conform to the laws of thought. 

Logic is mainly a formal science, having for its object 
to ascertain and describe all the general forms in which 
thought presents itself without regard to any subject-mat- 
ter. Logic differs from psychology in having for its ob- 
ject only the investigation of the formal laws of thought, 
while psychology treats of all the facts of the human mind 
and the laws by which its operations are guided. 



OUTLINES OF LOGIC. 



2. Utility of Logic. 

Logic does not teach us to think, but teaches us the 
laws by which our reasoning must be guided. All per- 
sons learn to think and to reason even before they know 
the name of logic, and thus unconsciously apply the prin- 
ciples of logic ; but many questions are of so complex and 
difficult a nature that it is only by the aid of logic that we 
are able to detect what is correct or fallacious in the argu- 
ment. The chief utility of logic thus consists in giving an 
invariable test of the correctness of an argument. 

3. Operations of the Mind. 

In approaching an argument the mind passes through 
the following intellectual processes : Perception, abstrac- 
tion, generalization, judgment, and reasoning. 

1. Perception is the act of the mind by which it gains 
knowledge of external objects through the senses. 

The products of perception are called percepts. Thus 
my idea of my house, or of Boston, or of any particular 
object, is a percept. 

2. Abstraction is the act of the mind by which it draws 
a quality away from an object and considers it apart from 
the other peculiarities of the object. 

Thus the observing of the color of a certain object and 
making that a distinct object of thought to the exclusion 



INTRODUCTION. 



of all the other qualities of the same object, is a process 
of abstraction. 

3. Generalization is the act of the mind by which it con- 
siders the qualities which are common to all the individuals 
of a group of objects and unites them into a single notion 
comprehending them. 

Thus if we consider the properties common to all kinds 
of triangles, disregarding difference in size or shape, the 
process is generalization. 

A concept or general notion is the product of abstraction 
and generalization. 

The concept plant, for instance, is formed by fixing our 
attention upon the properties common to all individual 
plants and disregarding all the points in which they differ. 
The concept plant thus embraces all individual plants, and 
is a name that may be applied to any one of them. 

A concept is always general, a percept particular. 

4. Judgment is the act of the mind by which we com- 
pare two objects of thought, asserting whether they agree 
or not. 

The product of this operation is called & judgment. A 
judgment expressed in words is called a proposition. 

5. Reasoning is the act of the mind which consists in 
drawing conclusions from two or more judgments. 

An act of reasoning in its simplest logical form is called 
a syllogism. 



CHAPTER I. 



FUNDAMENTAL LAWS OF THOUGHT. 

1. Definition of Law of Thought. 

A law of thought is a necessary and universal principle 
by which all thought must be governed. 

2. Fundamental Laws of Thought. 

There are four fundamental laws of thought, on which 
all reasoning must ultimately depend. These laws are: 

1. The Law of Identity (Principium identitatis). 

2. The Law of Contradiction (Principium contradictionis). 

3. The Law of Excluded Middle (Principium exclusi tertii). 

4. The Law of Sufficient Reason (Principium rationis 
snfficientis). 

1. The Law of Identity. — Whatever is, is. 

This law may be expressed by the formula A=A. Its 
meaning is that everything is identical with itself. All 
the attributes of a thing must be consistent with each 
other and with the thing itself. In the proposition All 

(4) 



FUNDAMENTAL LAWS OF THOUGHT. 



Americans are rational beings, the identity of All Ameri- 
cans with some rational beings is set forth. 

2. The Law of Contradiction. — Nothing can both be 
and not be. 

This law may be expressed by the formula A ?iot = 
not — A. The attributes of an object must not be incon- 
sistent with each other nor with the thing itself. In the 
proposition, No animals are plants, we assert that ani- 
mals are inconsistent with plants. A triangle may be 
either right-angled or not right-angled, but we cannot 
conceive that it should be both at the same time. If we 
say that a triangle is round, we evidently violate this law. 
because roundness is a quality inconsistent with a triangle. 

3. The Law of Excluded Middle. — Everything must 
either be or not be. 

This law may be expressed by the formula, A is either 
B or not — B. It is impossible to conceive of an}- thing 
and any quality without affirming that the quality either 
belongs to the thing or does not belong to it. Gold, for 
instance, must be either a metal or not a metal. There is 
no third. 

4. The Law of Sufficient Reason. — For every conse- 
quent there must be a sufficient reason. 

If two propositions are connected in such a manner 
that the truth of one necessarily implies the truth of the 



OUTLINES OF LOGIC. 



other, the former is called the reason and the latter the 
consequent. 

This law may be expressed bj the formula, If A is, B 
is. Its meaning is, that for every proposition that is not 
intuitively true a sufficient reason must be given. 

For instance, If two triangles have equal bases and 
equal altitudes, they are equivalent. Here the equivalence 
of the triangles is the consequent, and the reason why 
they are equivalent is that they have equal bases and 
equal altitudes. 



CHAPTER II. 



CONCEPTS. 



1. Definition of Concept. 



A concept or general notion is the consciousness in our 
mind of the attributes common to all the individuals of a 
certain group of objects. 

Concepts are formed by abstraction and generalization, 
as has already been mentioned. As examples of concepts 
we may give the following : Man, animal, book, triangle, 
plant, planet, heavenly body, and dog. 

2. Classification of Concepts. 

Concepts may be divided into 1. Positive and negative / 
2. Absolute and Relative j 3. Concrete and Abstract. 

1. a) A positive concept is one in which the existence 
of a quality is asserted. 

b) A negative concept is one in which the absence of 
a quality is asserted. 

(7) 



OUTLINES OF LOGIC. 



Thus, organic and right-angled are positive and inor- 
ganic and not right-angled negative concepts. 

2. a) An absolute concept is one that can be thought of 
without reference to some other concept. 

b) A relative concept is one that cannot be thought 
of without reference to some other concept. 

Thus, father, mother, son, and daughter are relative con- 
cepts. We cannot think of father or mother without ref- 
erence to a child, nor of son or daughter without reference 
to father or mother. Metal, water, and triangle, on the 
other hand, are terms which have no apparent relation to 
any other things, and which therefore are absolute. 

3. a) A concrete concept is a name that can be applied 
to a thing. 

b) An abstract concept is the name of a quality that 
belongs to a thing. 

Thus circle, table, and brick-house are concrete ; but 
redness, hardness, and usefulness abstract concepts. 

3. Content and Extent. 

Every concept has content and extent. 

By the content of a concept is meant all the marks or 
attributes of the concept. 

By the extent of a concept is meant all the individuals 
or objects it embraces. 

Let us take the concept insect. The content of insect 



CONCEPTS. 



consists of all the attributes which are necessarily pos- 
sessed by all insects and by which an insect is distin- 
guished from all other beings. By the extent of insect we 
mean all the different kinds of insects that exist. 

When we compare two concepts that are related to one 
another, we observe that the concept which is poorer in 
content has the greater extent, and that the one that has 
the greater content has the smaller extent ; or as it is 
usually expressed : 

The content and extent of two concepts are in inverS( 
ratio to each other. 

In order to make this clear let us compare the two terms 
fish and vertebrate. The term vertebrate includes not only 
all the animals that are included under the term fish, but 
also reptiles, birds, mammals, etc. Consequently vertebrate 
has a greater extent than fish. On the other hand, all the 
properties that belong to vertebrates must necessarily be- 
long to all fishes, and in addition to these there are many 
properties that belong exclusively to fishes and by which 
fishes are distinguished from all other vertebrates. There- 
fore fish, having a greater number of marks or attributes 
than vertebrate, lias the greater content. Vertebrate is a 
term that may be applied to all fishes, and fish is an indi- 
vidual case of vertebrate. As another example let us take 
the two terms plane figure and circle. Of these the former 
has obviously the greater extent and the latter the greater 
content. 



10 OUTLINES OF LOGIC. 

If two concepts are so related that one includes the 
other, as the concepts vertebrate and fish, the one that in- 
cludes the other is called the higher concept, and the one 
that is included in the other is called the lower concept. 
Thus vertebrate is the higher and fish the lower concept. 

4. Relation of Concepts. 

Two or more concepts may be compared : 1st, with re- 
spect to content ; and 2d, with respect to extent. In the 
first case they may be either compatible or incompatible. 
In the second case they may be either subordinate or co- 
ordinate. 

1. Compatible Concepts. — Two concepts are said to be 
compatible when they both can be affirmed of the same 
subject, or both are included in the content of the same 
concept. 

The two terms equilateral and 'right-angled, for instance, 
may both be affirmed of a square, and are consequently 
compatible. Large and heavy are also two compatible 
terms, because they may be affirmed of the same subject. 

2. Incompatible Concepts. — Two concepts are said to 
be incompatible when they cannot both be affirmed of the 
same subject, or are not included in the content of the same 
concept. 

Incompatible concepts are either contradictory or con- 
trary. 



CONCEPTS. 11 

a) Two concepts are contradictory when one is the nega- 
tive of the other. 

For instance, cold and not-cold, figure and not-figure, 
organic and inorganic, etc. From a logical point of view, 
it is immaterial which one of two contradictory terms is 
considered positive and which negative. Each is the neg- 
ative of the other. 

b) Two concepts are contrary when one not only implies 
a negation of the other, but also expresses some positive 
attribute. 

For instance, man and woman, pentagonal and hexago- 
nal. 

3. Subordinate Concepts. — Of two concepts, one is said 
to be subordinate to the other when it is included in the 
extent of the other. 

If one concept is included in the extent of another, the 
former is called sjjecies and the latter genus. Thus, of the 
two terms plant and tree, plant is the genus, and tree is a 
species of the genus plant. The genus has always a 
greater extent than the species, i. e., includes a greater 
number of individuals than the species. But as extent 
and content are in inverse ratio to each other, it follows 
that the species has greater content or a greater number 
of attributes than the genus. The species has not only all 
the attributes of the genus, but also other attributes by 
which it is distinguished from all other species of the same 



12 OUTLINES OF LOGIC. 

genus. The relation between two concepts of which one 
is subordinate to the other is shown by the diagram, where 
A (the outer circle) represents the genus and B (the inner 
circle) the species. 




4. Cookdinate Concepts. — Two or more concepts are 
said to he co-ordinate to each other, when they are included 
in the extent of the same concept, out at the same time ex- 
clude each other. 

Thus, the two concepts plant and animal are coordinate 
to each other, being both species of the same genus organic 
being, and also excluding each other. The terms plant 
and tree are not coordinate. They are both included in 
the extent of organic being, but they do not exclude each 
other. As another example of coordinate terms we may 
take Jish, bird, and mammal, all three being species of the 
genus vertebrate. The relation between coordinate terms 
is shown by the diagram, where A represents the genus 
and B, C, and D three of its species. 



CONCEPTS. 



13 




If one concept is subordinate to another, they must be 
compatible y and if two or more concepts are co-ordinate to 
each other, they must be incompatible. 

The truth of this may be verified by taking the terms 
parallelogram and quadrilateral and the terms plant and 
animal. Of the two terms parallelogram and quadrilat- 
eral, the former is subordinate to the latter. But as all 
rectangles are parallelograms, and also all rectangles are 
quadrilaterals, we see that the two terms parallelogram 
and quadrilateral may both be affirmed of the same sub- 
ject, rectangle. Hence parallelogram and quadrilateral are 
compatible. 

The two terms plant and animal are evidently coordi- 
nate to each other, both being species of the genus organic 
being, and at the same time excluding each other. But 
we cannot find any subject of which they may both be 
affirmed. Hence they are incompatible. 



CHAPTER III. 



JUDGMENTS. 

1. Definition of Judgment. 

Judgment is that act of thought by which we compare 
two objects of thought, asserting whether they agree or not. 

The product of this operation is called a judgment. A 
logical proposition is a judgment expressed in words. For 
instance, All horses are mammals. 

2. Terms. 

Every judgment contains two ideas, called the terms of 
the judgment. The term of which something is affirmed 
or denied is called the subject, and the term which is af- 
firmed or denied of the subject is called the predicate. 
The word that expresses the connection between the sub- 
ject and the predicate is called the copula. 

Thus, in the judgment 

Man is mortal, 
man is the subject, mortal the predicate, and is the copula. 

Of the two terms of a judgment the predicate is usually 

(14) 



judgments. 15 



a concept, and the subject may be either a concept or a 
percept. Thus, in the proposition Insects are animals, 
both the subject and the predicate are concepts. In the 
proposition Chicago is a city, the subject is a percept and 
the predicate a concept. Sometimes botli terms may be 
percepts, as in Chicago is not London. 

3. Classification of Judgments. 

Judgments are classified according to quality, quantity, 
relation, and modality. 

1. Quality. — According to quality judgments are di- 
vided into affirmative and negative. 

a) An affirmative judgment is one in which the predi- 
cate is affirmed of the subject. 

For instance, 

All horses are animals. 

b) A negative judgment is one in which the predicate is 
denied of the subject. 

For instance, 

No roses are animals. 

2. Quantity. — According to quantity judgments are 
divided into universal and particular. 

a) A universal judgment is one in v)hich the predicate 
is affirmed or denied of the subject in its whole extent. 
For instance, 

All men are mortal. 



16 OUTLINES OF LOGIC. 

b) A particular judgment is one in which the predicate 
is affirmed or denied of the subject only in part of its ex- 
tent. 

For instance, 

Some animals are insects. 
No triangles are circles. 

A judgment which has for its subject a singular term is 
sometimes called a singular judgment, as Alexander was a 
conqueror. All singular judgments, however, are univer- 
sal, since in such a judgment the predicate is evidently 
affirmed or denied of the whole of the subject. 

A proposition is said to be indefinite when it has no 
mark of quantity whatever, leaving it ambiguous whether 
it is universal or particular. In all such cases, however, 
the proper mark of quantity can be prefixed. Thus, the 
indefinite proposition Man is mortal means All men are 
mortal. 

The combination of difference in quality with difference 
in quantity gives rise to four classes of judgments : 
Universal affirmative. A. 
Universal negative. E. 
Particular affirmative. I. 
Particular negative. O. 

These four classes of judgments are designated by the 
letters A, E, I, and 0. It is easy to remember what 
kind of judgment each letter represents by observing that 
A and I are the first two vowels of the Latin word affirmo, 



JUDGMENTS. \1 



and E and O the vowels of nego. We give the following 
examples : 

All insects are animals. A. 

No men are gods. E. 

Some men are wise. I. 

Some men are not wise. 0. 

In passing from a particular affirmative to a particular 
negative judgment, we prefix not to the predicate. When 
we pass from a universal affirmative to a universal nega- 
tive judgment, however, this is not sufficient. In that 
case the negative adjective no must be prefixed to the 
subject. Let us take the universal affirmative judgment 
All men are rational. By prefixing not to the predicate 
we have All men are not rational, which may be particu- 
lar and may imply that some men may be rational. It is 
therefore not a complete negation of the universal affirm- 
ative judgment All men are not rational. Hence, in 
order to express a complete denial of the universal affirm- 
ative judgment we must prefix no to the subject. Thus, 
No men are rational. 

3. Relation. — According to relation judgments are di- 
vided into categorical, hypothetical, and disjunctive. 

a) A categorical judgment is one in which the predicate 
is unconditionally affirmed or denied of the subject. 

The simplest form of a categorical judgment is, 
S is P. 

—2 



18 OUTLINES OF LOGIC. 

For instance, 

All trees are plants. 

Some heavenly bodies are not planets. 

h) A hypothetical judgment is one in which the predi- 
cate is affirmed or denied of the subject conditionally. 
The simplest form of a hypothetical judgment is, 
If A is B, C is D. 

A hypothetical judgment thus consists of two categor- 
ical judgments connected by the conjunction if The 
first, or the one that expresses the condition, is called the 
antecedent, and the other the consequent. 

For instance, 

If rain does not come, the crops will fail. 

Here, If rain does not come is the antecedent, and the 
crops will fail is the consequent. 

A hypothetical judgment can always be changed to a 
categorical judgment of exactly the same meaning, having 
for its subject the antecedent and for its predicate the 
consequent of the hypothetical judgment. 

Thus, the hypothetical judgment 

If a triangle is equilateral, it is equiangular 
can be converted into the categorical judgment 
All equilateral triangles are equiangular. 

c) A disjunctive judgment is one that expresses an 
alternative. 



JUDGMENTS. 10 



The simplest form of a disjunctive judgment is 
S is either P or not P. 

The disjunctive judgment has instead of a single predi- 
cate two alternatives or more, of which one must be 
asserted of the subject to the exclusion of any other alter- 
native. For instance, 

John is either in the house or not in the house. 

This triangle is either right-angled, obtuse-angled, or 
acute-angled. 

• A disjunctive judgment is called divisive when the predi- 
cate expresses all the species of the subject. For instance, 
Organic beings are divided into animals and plants. 
Triangles are divided into right-angled and oblique- 
angled. 
The divisive judgment is disjunctive only in form, but 
categorical in sense. It is, in reality, composed of two or 
more particular judgments. Thus, the judgment Triangles 
are divided into right-angled end oblique-angled is com- 
posed of the two particular judgments 
Some triangles are right-angled. 
Some triangles are oblique-angled. 

4. Modality. — According to modality, or the degree of 
certainty, judgments are divided into apodictic, problematic, 
and assertory. 

a) An apodictic judgment is one which expresses the 



20 OUTLINES OF LOGIC. 

combination between the subject and the predicate as a 
necessity. 

S must be P. 
For instance, 

An equilateral triangle must be equiangular. 

b) A problematic judgment is one which expresses the 
combination between the subject and the predicate as a pos- 
sibility. 

S may be P. 
For instance, 

Mars may be inhabited. 

c) An assertory judgment is one which expresses th& 

combination between the subject and the predicate as a fact 

to be taken for granted. 

8 is P. 
For instance, 

This dog is mad. 

4. Distribution of Terms. 

A term is said to be distributed when it is taken uni- 
versally or in its whole extent. 

For instance, in the judgment All animals are organic 
beings, the term animal is taken universally or in its whole 
extent, and is therefore distributed. 

1. With regard to the subject we have the following 
rules : 

a) In All S are P (A) 

and No S are P (E) 
the subject is distributed, both judgments being universal. 



JUDGMENTS. 21 



b) In Some S are P (I) 

and Some S are not P (0) 
the subject is not distributed, both judgments being partic- 
ular. 

2. With regard to the predicate we have the following 
rules : 

a) In All S are P (A) 

the predicate is not distributed. It is evident that the 
whole of P is not considered, as P may contain many 
other things besides S. 

b) In Some S are P (l) 

the predicate is not distributed, as is shown by the same 
reasoning as for A. 

c) In No S are P (E) 

the predicate is distributed. In order to assert that no 
part of S belongs to any part of P, it is evident that the 
whole of P must be considered. 

d) In Some S are not P (O) 

the predicate is distributed. The same reasoning applies 
here as for E. We must consider the whole of P in or- 
der to assert that no part of it belongs to some P in ques- 
tion. 

REMARKS. 

1. A distributes the predicate in case the subject and 
the predicate are co-extensive, i. e., have exactly the same 
extent. 

For instance, 

All equilateral triangles are equiangular. 



22 



OUTLINES OF LOG*/C. 



2. / distributes the predicate in case the subject is the 

genus and the predicate one of its species. For instance, 
Some animals are vertebrates. 

For the distribution of terms in the four categorical judg- 
ments we have then the following rules : 

1. Universal judgments distribute the subject/ 'par- 

ticular judgments do not. 

2. Negative judgments distribute the predicate/ af- 

firmative judgments do not. 
These rules may be stated by the following schedule : 



SUBJECT. 

A. Distributed. 
E. Distributed. 
I. Undistributed. 
O. Undistributed. 



PREDICATE. 

Undistributed. 
Distributed. 
Undistributed. 
Distributed. 



In the diagrams given below the distribution of the sub- 
ject and the predicate in the four categorical judgments is 
shown. S represents the subject and P the predicate. 
A. K 






JUDGMENTS. 23 



/. 





5. Immediate Inference. 

Immediate inference is that act of thought by which we 
transform one judgment into another and from the validity 
or invalidity of one infer the validity or invalidity of the 
other. 

We will treat immediate inference under the following 
heads : 

1. Synonymous Judgments. 

2. Subalterns. 

3. Opposition. 

4. Conversion. 

1. Synonymous Judgments. — Two judgments are synon- 
ymous when they express the same fact in different words. 

The wording of a proposition may evidently be changed 
in many different ways so as to give a new proposition, 
differing only in form but not in sense from the given 



24 OUTLINES OF LOGIC. 

one. We may, for instance, substitute for either the sub- 
ject or the predicate equivalent terms ; or change from a 
categorical to a hypothetical proposition, and conversely ; 
or instead of affirming one thing, deny its opposite. Evi- 
dently both are true or both false at the same time. 
For instance : 

(True) This is a triangle. 

(True) This is a figure having three sides. 

(False) All triangles are equilateral. 
(False) No triangles are not equilateral. 

(True) Damp gunpowder will not explode. 
(True) If gunpowder is damp, it will not explode. 

2. Subalterns. — Two judgments are said to he subalterns 
when they have the same subject, the same predicate, and 
the same quality, but one is universal and the other partic- 
ular. 

Thus, A and / are a pair of subalterns ; also E and 0. 
I and O are called the subalternates of A and E respect- 
ively, each of which is a subalternans. 

From the truth of the 'universal we infer the truth of 
the 'particular, and from the falsity of the particular we 
infer the falsity of the universal. But the truth of the 
particular does not always include the truth of the uni- 
versal / nor does the falsity of the universal always include 
the falsity of the particular. 



JUDGMENTS. 25 



For instance, 

(True) All men are mortal. (A) 
(True) Some men are mortal. (/) 

(True) No animal is rational. (E) 
(True) Some animals are not rational. (O) 

(False) Some plants are animals. (/) 
(False) All plants are animals. (A) 

(False) Some triangles are not figures. (O) 
(False) No triangles are figures. (E) 

But the truth of the particular judgment 
Some animals are insects (/) 
does not involve the truth of the universal 
All animals are insects. (A) 
Nor can we from the falsity of the universal judgment 

No figures are triangles (E) 
infer the falsity of the particular 

Some figures are not triangles. (#) 
Hence we conclude from the truth of A and E to the 
truth of / and O respectively, and from the falsity of / 
and to the falsity of A and E respectively; but not 
from the falsity of A and E to the falsity of / and re- 
spectively, nor from the truth of / and O to the truth of 
A and E respectively. 

3. Opposition. — Opposition takes place between two 
judgments when they have the same subject and the same 
predicate, but opposite quality. 



26 OUTLINES OF LOGIC. 

There are three kinds of opposition depending on the 
quantity of the judgments, viz., contrary, contradictory, 
and subcoyitrary . 

a) Contrary. — If both judgments are universal, the op- 
position is said to be contrary, or the judgments are con- 
traries each of the other. 

Two contrary judgments cannot both be true, but they 
may both be false. 

Hence the truth of one involves the falsity of the other, 
hut the falsity of one does not necessarily involve the truth 
of the other. 

For instance, 

(True) All trees are plants. (A) 
(False) No trees are plants. <E) 
(True) No animals are plants. (E) 
(False) All animals are plants. (A) 
But from the falsity of 

All animals are insects (A) 
we cannot infer the truth of 

No animals are insects. (E) 
Nor can we from the falsity of 

No animals are birds (E) 
infer the truth of 

All animals are birds. (A) 

Hence we conclude from the truth of A to the falsity of 
E and from the truth of E to the falsity of A, but not 



JUDGMENTS. 27 



from the falsity of A to the truth of E, nor from the 
falsity of E to the truth of A. 

b) Contradictory. — If one judgment is universal and 
the other particular, the opposition is said to be contradic- 
tory, or the judgments are contradictories each of the other. 

Of two contradictory judgments one must be true and 
the other false. 

Hence the truth of one involves the falsity of the other, 
and the falsity of one involves the truth of the other. 
For instance, 

(True) All plants are organic beings. (A) 
(False) Some plants are not organic beings. (O) 
(True) No triangles are squares. (E) 
(False) Some triangles are squares. (/) 
(True) Some animals are not birds. (6>) 
(False) All animals are birds. [A) 
(True) Some plants are water-plants. (/) 
(False) No plants are water-plants. (J?) 

Hence we conclude from the truth or falsity of A, E, I, 
and O to the falsity or truth of O, I, E, and A respect- 
ively. 

c) Subcontrary. — If both judgments are particular, the 
opposition is said to be subcontrary, or the judgments are 
subcontraries each of the other. 

Two subcontrary judgments may both be true, but they 
cannot both be false. 



28 OUTLINES OF LOGIC. 

Hence from the falsity of one we infer the truth of the 
other, hut the truth of one does not necessarily involve the 
falsity of the other. 

For instance, 

(False) Some triangles are not figures. (0) 
( True) Some triangles are figures. (/) 

(False) Some plants are animals. (/) 
(True) Some plants are not animals. (0) 

But from the truth of 

Some heavenly bodies are planets (/) 
we cannot infer the falsity of 

Some heavenly bodies are not planets. [0) 
Nor can we from the truth of 

Some animals are not fishes (0) 
infer the falsity of 

Some animals are fishes. (/) 

Hence we conclude from the falsity of and I to the 
truth of / and respectively, but not from the truth of O 
and / to the falsity of / and respectively. 



JUDGMENTS. 



29 



The relations between the four judgments A, £", 1, and 
are shown by the following schedule: 




4. Conversion. — A judgment is said to undergo conver- 
sion or to be converted token its subject and predicate are 
interchanged. 

If the given judgment is true, the new judgment must 
also be true. 



30 



OUTLINES OF LOGIC. 



There are three kinds of conversion : simple conversion, 
conversion by limitation, and conversion by contraposition. 

a) Simple conversion. — A judgment is simply converted 
when its subject and predicate are interchanged, the qual- 
ity and quantity remaining the same. 

For instance, 

jO \ No metals are compounds. (E) 
No compounds are metals. (E 1 ) 




Some flowers are yellow. (/) 
Some yellow things are flowers. (/) 



But from the judgment 




All metals are elements [A) 
we cannot infer that 
All elements are metals. 



Nor can we pass from 




Some plants are not water-plants (O) 

to 

Some water-plants are not plants. 



Hence only universal negative and particular affirma- 
tive judgments can be simply converted. 



JUDGMENTS. %\ 



b) Conversion by limitation. — A judgment is said to be 
converted by limitation when its subject and predicate are 
interchanged, the quality remaining the same, but the 
quantity being changed. 

For instance, 




All men are mortal. (^4) 

Some mortal beings are men. (7) 



But from the judgment 




Some animals are not insects (O) 

we cannot pass to 

No insects are animals. 



Hence all universal affirmative judgments can be con- 
verted by limitation. To particular negative judgments 
neither simple conversion nor conversion by limitation 
can be applied. 

There are, however, some universal affirmative judg- 
ments that can be simply converted ; namely, all those in 
which the subject and the predicate are co-extensive. To 
that class belong all logical definitions. 

For instance, 

A quadrilateral is a figure having four sides. 
All figures having four sides are quadrilaterals. 



32 



OUTLINES OF LOGIC. 



All equilateral triangles are equiangular. 
All equiangular triangles are equilateral. 



c) Conversion by contraposition. — We are said to con- 
vert a judgment by contraposition when we first change 
the quality and for the predicate substitute its contradic- 
tory and then apply simple conversion. 

By the first process we pass from the affirmation of one 
thing to the denial of its opposite. For instance, 



All metals are elements (A) 
No metals are not-elements (E) 



and then by simple conversion 

No not-elements are metals (JE) 





Some animals are not insects (0) 
Some animals are not-insects (/) 



and by simple conversion 

Some not-insects are animals (/). 

In the particular negative judgment we thus simply 
transfer the negative particle from the copula to the pred- 
icate and then apply simple conversion. 



JUDGMENTS. 33 



Hence all universal affirmative and particular negative 
judgments can be converted by contraposition. 

A similar process may be applied to the universal neg- 
ative judgment, though in that case we can only convert 
by limitation. For instance, 




No fishes are birds. (E) 
All fishes are not-birds. (A) 



and by conversion by limitation 

Some not-birds are fishes. (/) 

For conversion we have then the following rules : 

I. Only universal negative and particular affirmative 
judgments can be simply converted. 

II. All universal affirmative judgments can be converted 
by limitation. 

III. Particular negative judgments can only be converted 
by contraposition. 

6. Simple and Complex Propositions. 

A simple proposition is one that has only one subject 
and one predicate. For instance, 

Gold is a metal. 
A complex proposition is one that has more than one 

-3 



34 OUTLINES OF LOGIC. 

subject, or more than one predicate, or both. For in- 
stance, 

Birds and fishes are animals. 

In this example there are evidently two categorical prop- 
ositions combined in one, viz., 

Birds are animals 
and Fishes are animals. 

The only complex propositions with which logic is di- 
rectly concerned are the hypothetical and the disjunctive 
propositions, which have already been described. 



CHAPTER IV. 



SYLLOGISMS. 
1. Definition of Syllogism. 

Syllogism is the process by which two objects of thought 
are compared through their relation to a third. 

Every syllogism contains three terms, the major term, 
the middle term, and the minor term. The relation be- 
tween the three terms is expressed by three judgments, of 
which two are called the premises and the third the conclu- 
sion. In one premise the middle term is compared with 
the major term, in the other premise it is compared with 
the minor term, and in the conclusion the major and 
minor terms are compared. The premise containing the 
major term is called the major premise, and the premise 
containing the minor term is called the minor premise. 
The middle term, being only the medium of comparison 
between the two other terms, occurs only in the premises, 
but not in the conclusion. The minor term is always the 
subject of the conclusion, and the major term is always the 
predicate of the conclusion. 

(35) 



36 OUTLINES OF LOGIC. 

The minor and major terms are so called because the 
major term has usually greater extent than the minor 
term. The three terms of a syllogism are usually repre- 
sented by the letters P, M, and S. P designates the 
major term, being the predicate of the conclusion ; M de- 
notes the middle term ; and S denotes the minor term, 
being the subject of the conclusion. 

The three judgments of a syllogism are usually arranged 
in the following order : 

Major premise. All men are rational. 
Minor premise. All Americans are men. 
Conclusion. All Americans are rational. 

In the example given above, men is the middle term, 
rational the major term, and Americans the minor term. 

The syllogism may also be defined as the act of thought 
by which from two given judgments, called the premises, 
we draw or infer a third judgment, called the conclusion. 
Syllogism is also called mediate inference, and differs from 
immediate inference, described in the preceding chapter, 
mediate inference being made through a medium or a 
middle term. 

2. Classification of Syllogisms. 

Syllogisms are divided into categorical, hypothetical, and 
disjunctive. 

1. A categorical syllogism is a syllogism having for its 
major premise a categorical judgment. 



SYLLOGISMS. 



37 



2. A hypothetical syllogism is a syllogism having for 
its major premise a hypothetical judgment. 

3. A disjunctive syllogism is a syllogism having for its 
major premise a disjunctive judgment. 

Examples : 



Categorical. 



Hypothetical. 



Disjunctive. 



M is P. 

S is M. 

S is P. 

If A is B, C is D. 

A is B. 

CisD. 

A is either B or not-B. 

A is B. 

A is not not- B. 



3. Categorical Syllogisms. 

1. Definition. — A categorical syllogism is a syllogism 
having for its major premise a categorical judgment. 

The minor premise and the conclusion are also categor- 
ical judgments. 

2. Rules. — A general rule for the syllogism is an axiom 
known as the dictum de omni et nullo of Aristotle. This 
axiom may be stated thus : 

Whatever is affirmed or denied of a whole class may also 
be affirmed or denied of any individual contained in that 
class. 



38 OUTLINES OF LOGIC. 

The special rules of the categorical syllogism are : 

I. The syllogism must contain three and only three terms. 

II. The syllogism must contain three and only three 
judgments. 

III. The middle term must he distributed at least in one 
of the premises. 

IV. In order that a term may be distributed in the con- 
clusion it must be distributed in one of the premises. 

V. From two negative premises no conclusion can be 
drawn. 

VI. From two particular premises no conclusion can be 
drawn. 

VII. If one premise is negative the conclusion will be 
negative. 

VIII. If one premise is particidar the conclusion will 
be particular. 

3. Explanation of the Rules. — The first and second 
rules need no further explanation. 

3d rule. If the middle term were not distributed in at 
least one of the premises, it might happen that the minor 
and major terms are compared with different parts of the 
middle term, and therefore the middle term would no 
longer be a medium of comparison. For instance, 
All P are M 
All S are M. 

Here the middle term is not distributed. P is one part 



SYLLOGISMS. 39 



of M and S is another part of M, and these parts may or 
may not coincide. No relation can be established between 
S and P, as S may fall wholly without, or wholly within, 
or partly without and partly within P, as is seen in the di- 
agram. 




Jf.th rule. If either the major or minor term is not dis- 
tributed in the premise where it occurs, it must not be 
distributed in the conclusion. It is evident that we are 
only enabled to infer something about that part of either 
the major or minor term which has been compared with 
the middle term in the premise. In the syllogism 
All insects are animals 
No dogs are insects 
,\ No dogs are animals 

the major term animals is not distributed in the major 
premise, but is distributed in the conclusion. This argu- 
ment is consequently fallacious. This fallacy is called an 
illicit process of the major terra. 



40 OUTLINES OF LOGIC. 

Again, in the example 

All flies are insects 

All flies are animals 
.*. All animals are insects 
the minor term animals is distributed in the conclusion, 
but not in the minor premise. Hence the argument is 
false. This kind of fallacy is called an illicit process of 
the minor term. 

5th rule. If both premises are negative, no conclusion 
can be drawn, because the middle is no longer a medium 
of comparison between the minor and major terms. For 
instance, from the premises 

No M are P 

No S are M 
no conclusion can be drawn as regards the relation be- 
tween S and P, as S may fall wholly within, or wholly 
without, or partly within and partly without P, as is seen 
in the diagram. 




SYLLOGISMS. 41 



6th rule. If both premises are particular, no conclusion 
can be drawn, because no relation can be established be- 
tween two terms that are only partly connected with a 
third. From the premises 

Some M are P 

Some S are M 
no conclusion can be drawn ; for, as is shown by the dia- 
gram, S may fall wholly without, or wholly within, or 
partly without and partly within P. 




7th rule. If the minor premise, for instance, be negative, 
thus expressing a disagreement between the minor and 
middle terms, and the major premise affirmative, express- 
ing an agreement between the major and middle terms, 
the conclusion must necessarily express a disagreement 
between the minor and major terms, i. e., the conclusion 
must be negative. And in the same way if the major 
premise is negative and the minor premise affirmative. 

8th rule. This rule is in fact a corollary of the third and 
fourth rules. 



42 



OUTLINES OF LOGIC. 



4. Figures. — The three terras of a syllogism may be 
arranged in different ways. It is evident that the middle 
term can have only four different positions, and hence 
there are four different ways, or, as they are called, figures 
in which the terms of a syllogism may be arranged. 
These four figures of the syllogism are shown in the fol- 
lowing scheme : 



1 


2 


3 


4 


M is P. 

SisM. 


Pis M. 

SisM. 


M is P. 

Mis S. 


PisM. 

MisS. 


Sis P. 


S is P. 


Sis P. 


S is P. 



In the first figure the middle term is the subject of the 
major premise and the predicate of the minor premise. 

In the second figure the middle term is the predicate of 
both the major and the minor premises. 

In the third figure the middle term is the subject of 
both the major and the minor premises. 

In the fourth figure the middle term is the predicate of 
the major premise and the subject of the minor premise. 

The first three figures were proposed by Aristotle, and 
hence they are usually called the Aristotelian figures. The 
fourth figure, proposed by Galen, is really an inversion of 
the first figure and is comparatively useless, because the 
same conclusions can be obtained more naturally by using 
the first figure. 



SYLLOGISMS. 43 



5. Moods. — As every syllogism must contain two prem- 
ises and each premise may be either universaVaffirmative, 
universal negative, particular affirmative) or particular 
negative, there would be in each figure sixteen different 
forms of the syllogisms, or, as they are called, moods, de- 
pending on the quality and quantity of the premises. But 
the number of moods in each figure is limited by the rules 
of the syllogism mentioned above, and thus omitting all 
moods which violate these rules and all moods which are 
useless, being included in other moods, there will remain 
only nineteen. As an artificial aid in memorizing these 
nineteen possible moods the following mnemonic verses 
have been invented : 

Fig. 1. Barbara, Celarent, Darii, Ferioque, prions ; 

Fig. 2. Cesare, Carnestres, Festino, Baroko, secundae ; 

Fig. 3. Tertia, Darapti, Dlsamis, Datisi, Felapton, Bo- 
kardOj Ferison, habet ; quarta insuper addit, 

Fig. 4. Brainantip, Camenes, Dunaris, Fesapo, Fresison. 

Each of the italicized names represents a mood, the 
vowels of each name standing for the three judgments of 
the syllogism. Thus for instance Cesare signifies the 
mood of the second figure, which has E for the major 
premise, A for the minor, and E for the conclusion. 

First Figure. 
This is the only figure in which the conclusion can be 



4:4: 



OUTLINES OF LOGIC. 



universal affirmative. With regard to the premises the 
following rules will be observed : 

d) The major premise must he universal. 

b) The minor premise must be affirmative. 

These special rules can easily be deduced from the gen- 
eral rules of the syllogism. 

The four valid moods in this figure are : 

Barbara, Celarent, Darii, and Ferio. 



Barbara. 



EXAMPLES. 




All men are mortal. 
All Americans are men. 
,\ All Americans are mortal. 



Celarent. 




Darii. 




No quadrilaterals are cir- 
cles. 

All parallelograms are 
quadrilaterals. 
.'.No parallelograms are cir- 
cles. 



All mammals have red blood. 
Some animals are mammals. 
.*, Some animals have red blood. 



SYLLOGISMS. 



45 



Ferio. 




No insects are warm-blooded. 

Some animals are insects. 

,*, Some animals are not warm- 
blooded. 



Second figure. 

In this figure the conclusion is always negative. For 
the premises we have the following special rules : 

a) The major premise must he universal . 

b) One of the premises must be negative. 
The four valid moods of this figure are : 

Cesare^ Camedres, Festino, and Baroko. 



EXAMPLES. 



Cesar e. 




o... 



Camestres. 





No trapezoid is equilat- 
eral. 

All squares are equilat- 
eral. 

No squares are trape- 
zoids. 



All men are rational. 
/U \ No apes are rational. 
,\ No apes are men. 



46 



OUTLINES OF LOGIC. 



Festino. 




Baroko. 



No planets are self-luminous. 
Some heavenly bodies are 

self-luminous. 
.*. Some heavenly bodies are not 

planets. 



All horses are mammals. 
Some animals are not mammals. 
,•. Some animals are not horses. 



Third Figure. 

In this figure the conclusion is always particular. For 
the premises we have the following rule : 

The minor premise must he affirmative. 

Six moods are possible, viz. : 
Darapti, Disamis, Datlsi, Felapton, Bokardo, and Ferison. 




Darapti. 



EXAMPLES. 




All whales are mammals. 
All whales live in water. 
.'. Some animals living in wa- 
ter are mammals. 



SYLLOGISMS. 



47 



Disarms. 




Datisi. 




Felapton. 




Bol'drdo. 



Some parallelograms are rectangles. 
All parallelograms are quadrilaterals. 
Some quadrilaterals are rectangles. 



All parallelograms are quadri- 
laterals. 

Some parallelograms are equi- 
lateral. 
.*. Some equilateral figures are 
quadrilaterals. 



No man is omniscient. 
All men are rational. 
Some rational beings are not omnis- 
cient. 




Some plants are not trees. 
All plants are living beings. 
Some living beings are not trees. 



48 



OUTLINES OF LOGIC. 



Ferison. 




No animals are plants. 
Some animals live in water. 
.'. Some organisms living in water are 
not plants. 



Fourth Figure. 
In this figure there are five valid moods, viz. : 
Bramantip, Oamenes, Dimaris, Fesapo, and Fresison. 



Bramantip. 




Camenes. 




EXAMPLES. 

All fishes breathe by gills. 

All animals breathing by gills are 

cold-blooded. 
Some cold - blooded animals are 

fishes. 



All men are mortal. 

No mortal being is om- 
niscient. 
,\ No omniscient being is a 
man. 



SYLLOGISMS. 



49 



Di maris. 




Fesapo. 




Fresison. 




Some taxes are oppressive. 
All oppressive things should be 
repealed. 
,\ Some things which should be re- 
pealed are taxes. 



No immoral acts are proper amuse- 
ments. 

All proper amusements are de- 
signed to give pleasure. 

Some things designed to give pleas- 
ure are not immoral acts. 



No birds have gills. 

Some animals having gills 

are vertebrates. 
Some vertebrates are not 

birds. 



4. Hypothetical Syllogisms. 

1. Definition. — A hypothetical syllogism is a syllogism 
which has for its major premise a hypothetical judgment. 

The minor premise and the conclusion are usually cate- 
gorical judgments. If all three judgments are hypothet- 



50 OUTLINES OF LOGIC. 

ical, the syllogism follows the same rules as the categorical 
syllogism, to which it can easily be reduced. 

For instance, 

If a man violates the laws, he ought to be punished. 
If a man commits murder, he violates the laws. 
,\ If a man commits murder, he ought to be punished. 

This can easily be put in the form of a categorical syl- 
logism as follows : 

A man that violates the laws ought to be punished. 
A murderer violates the laws. 
,\ A murderer ought to be punished. 

In the following we will therefore only consider hypo- 
thetical syllogisms in which the minor premise and the 
conclusion are categorical judgments. 

2. Moods. — Hypothetical syllogisms are divided into 
constructive and destructive, according as the minor prem- 
ise is affirmative or negative. The first form is also called 
the modus ponens, or the mood that affirms, and the sec- 
ond the modus tollens, or the mood that denies. 

a) Modus ponens. — For this mood we have the follow- 
ing rule : 

If the antecedent he affirmed, the consequent must he 
affirmed. The minor premise a firms the antecedent and 
the conclusion affirms the consequent. 



SYLLOGISMS. 51 



The general form of a constructive hypothetical syl- 
logism is 

If A is B, Cis D. 
A is B. 

.-. C is D. 

For instance, 

If a triangle is equilateral, it is equiangular. 
This triangle is equilateral. 
,*. This triangle is equiangular. 

If he has a fever, he is sick. 
He has a fever. 
,\ He is sick. 

b) Modus toll ens. — For this mood we have the follow- 
ing rule : 

If the consequent be denied, the antecedent must be de- 
nied. The minor premise denies the consequent, and the 
conclusion denies the antecedent. 

The general form of a destructive hypothetical syl- 
logism is 

If A is B, C is D. 

C is not D. 
,\ A is not B. 
For instance, 

If a triangle is equilateral, it is equiangular. 
This triangle is not equiangular. 
,\ This triangle is not equilateral. 



52 OUTLINES OF LOGIC. 

If a man is a murderer, he ought to be punished. 
This man ought not to be punished. 
.'. This man is not a murderer. 

3. Fallacies. — If the minor premise either affirms the 
consequent or denies the antecedent, a fallacy of argument 
arises. If we affirm the consequent, we may not therefore 
affirm the antecedent, because the consequent might follow 
from some other antecedent as well as from the one given ; 
or, as we might express it, a given effect may be produced 
by several different causes. For the same reason it is evi- 
dent that we cannot pass from the denial of the antecedent 
to the denial of the consequent. Thus the argument, 

If he has a fever, he is sick. 

He is sick. 
.*. He has a fever. 
is fallacious. If a person is sick, it does not necessarily 
follow that he has a fever. He may be sick from some 
other cause. For the same reason the argument, 

If he has a fever he is sick. 

He has not a fever. 
.*. He is not sick, 
is fallacious. 

There is one exception to this rule, and that is in case 
the given condition is the only condition of the consequent. 
In such a case we may pass from the affirmation of the 
consequent to the affirmation of the antecedent, or from 



SYLLOGISMS. 53 



the denial of the antecedent to the denial of the conse- 
quent. For instance, 

If a triangle is equilateral, it is equiangular. 

This triangle is equiangular. 
.*, This triangle is equilateral. 

If a triangle is equilateral, it is equiangular. 
This triangle is not equilateral. 
.'. This triangle is not equiangular. 

In the above examples the two terms equilateral tri- 
angle and equiangular triangle are evidently co-extensive. 

4. Reduction of Hypothetical to Categorical Syl- 
logisms. — As we have already seen, every hypothetical 
judgment can be converted into a universal affirmative 
judgment. Hence every hypothetical syllogism can be 
reduced to the categorical form and will consequently fol- 
low the rules laid down for the categorical syllogisms. 
In order to illustrate this we take the following example : 
If an animal is a mammal, it has red blood. 
All horses are mammals. 
.*, All animals have red blood 
By changing the major premise into a categorical judg- 
ment we obtain a categorical syllogism in the mood B<ir 

bara. 

All mammals have red blood. 

All horses are mammals. 

.*. All horses have red blood. 



54 OUTLINES OF LOGIC. 

5. Disjunctive Syllogisms. 

1. Definition. — A disjunctive syllogism is a syllogism 
which has for its m,ajor premise a disjunctive judgment. 

The minor premise and the conclusion are categorical 
judgments. 

2. Rules. — The general rule governing all disjunctive 
syllogisms is : 

If one or 'more alternatives he affirmed, the rest must be 
denied, and if one or more alternatives be denied, the rest 
must be affirmed. 

This rule follows immediately from the law of excluded 
7niddle. 

3. Moods. — There are two moods, viz., modus ponendo 
tollens (the mood which by affirming denies) and modus 
tollendo ponens (the mood which by denying affirms), ac- 
cording as the minor premise is affirmative or negative. 

a) Modus ponendo tollens. — The general form of this 
mood is A is either B or not-B. 

A is B. 
.'. A is not not-B. 
For instance, 

A triangle is either right-angled, acute-angled, or 

obtuse-angled. 
This triangle is right-angled. 
.*. This triangle is neither acute -angled nor obtuse- 
angled. 



SYLLOGISMS. 55 



b) Modus tollendo ponens. — The general form of this 

mood is 

A is either B or not-B. 

A is not not-B. 

.-. A is B. 

For instance, 

A triangle is either right-angled, acute-angled, or 

obtuse-angied. 
This triangle is neither right-angled nor acute- 
angled. 
.•. This triangle is obtuse angled. 

6. Dilemma. 

A dilemma is a syllogism having for its major premise 
a hypothetical judgment and for its minor premise a dis- 
jun ctive j udgment. 

There are several different forms of the dilemma. We 
will only give one of the more common forms, in which 
the major premise is a hypothetical judgment whose con- 
sequent is disjunctive. This form of the dilemma may be 

stated thus : 

If A is, either B or C is. 

Now neither B nor C is. 

/. A is not. 

This is in fact a destructive hypothetical syllogism. All 



56 



OUTLINES OF LOGIC. 



possible alternatives of the consequent are denied, there- 
fore the antecedent must also be denied. For instance, 

If this triangle is not right-angled, it must be either 

obtuse-angled or acute-angled. 
Now it is neither obtuse-angled nor acute-angled. 
.*. It must be right-angled. 



7. Compound Syllogisms. 

A series of syllogisms combined together in such a man- 
ner that the conclusion of the first is taken as a premise 
of the second and so on is called a compounU syllogism or 
a poly-syllogism. 

When the conclusion of one syllogism is used as a 
premise of another syllogism, the former syllogism is 
called a pro-syllogism and the latter an epi-syllogism. The 
conclusion of a pro-syllogism may be either the major or 
the minor premise of the epi-syllogism, as is seen by the 
following examples : 

1. 2. 

All C are D. / "\ All B are C. 

All B are C. / /s' ^\\ A AU A are B ' 

.-, All B are D. / / f/^ x\ \ _\ .-. All A are C. 



All B are D. 
All A are B. 
All A are D. 




All C are D. 
All A are C. 

.-. All A are D. 



SYLLOGISMS. 57 



For A, B, C, and D let us take the terms square, 'paral- 
lelogram, quadrilateral, and figure, and we have the fol- 
lowing compound syllogisms : 

1. 

All quadrilaterals are figures. 
All parallelograms are quadrilaterals. 
.*, All parallelograms are figures. 

All parallelograms are figures. 
All squares are parallelograms. 
.•. All squares are figures. 

2. 
All parallelograms are quadrilaterals. 
All squares are parallelograms. 
.*. All squares are quadrilaterals. 

All quadrilaterals are figures. 
All squares are quadrilaterals. 
.•. All squares are figures. 

8. Abridged Syllogisms. 

An abridged syllogism is a syllogism [either simple or 
compound) in which one or more of the premises is sup- 
pressed. 

This is the usual form of an argument. Perfectly 
formal syllogisms are very seldom met with. But in 
order that an argument which has not the form of a per- 
fect syllogism may be valid it must be capable of being 



58 OUTLINES OF LOGIC. 

put into the form of regular syllogisms. It should also 
be observed that, though one or more premises may be 
suppressed, no term must be wanting. 

The different kinds of abridged syllogisms which we 
will consider are : 

1. The Enthymeme. 

2. The Ejpichirema. 

3. The Sorites. 

1. Enthymeme. — An enthymeme is an abridged simple 
syllogism in which one or both of the premises is stip- 
pressed. 

The ethymeme is of two kinds. 

a) Either the major or the minor premise is suppressed. 
For instance, 

The square is a parallelogram. 
,\ The opposite angles of a square are equal. 

All men are mortal. 
,\ Napoleon is mortal. 
In the first example the major premise, and in the sec- 
ond the minor premise is suppressed. 

b) Both premises are suppressed, the middle term being 
included in the conclusion. For instance, 

The square, being a parallelogram, has the opposite 
sides equal. 
The enthymeme has very often the form of a sentence 
consisting of two propositions, united by the conjunction 



SYLLOGISMS. 59 



because. Thus, Napoleon is mortal because he is a man is 
really an enthymeme. It can easily be put into the form 
given above. 

2. Epichirema. — An epichirema is an abridged com- 
pound syllogism is which one or more of the premises are 
enthymemes. 

For instance, 

All minerals, being material bodies, have weight. 
Gold, being a metal, is a mineral. 
,\ Gold has weight. 
This may be put into the regular syllogistic form as fol- 
lows : 

All material bodies have weight. 

1. All minerals are material bodies. 
.*. All minerals have weight. 

All metals are minerals. 

2. Gold is a metal. 
.•. Gold is a mineral. 

All minerals have weight. 

3. Gold is a mineral. 
,\ Gold has weight. 

3. Sorites. — A sorites or chain-argument is an abridged 
poly-syllogism consisting of three or more simple premises. 

There are two kinds of sorites, the Aristotelian and the 
Goclenian, the former having been invented by Aristotle 



60 



OUTLINES OF LOGIC. 



and the latter by Goclenius. These two kinds of sorites 
may be stated in the following way : 



Aristotelian. 

All A are B. 
All B are C. 
All C are D. 
All D are E. 
\ All A are E. 




Goclenian. 

All D are E. 
All C are D. 
All B are C. 
All A are B. 
.-. All A are E. 



In the Aristotelian sorites the predicate of one premise 
becomes the subject of the next, and the conclusion has 
for its subject the subject of the first premise, and for its 
predicate the predicate of the last premise. 

In the Goclenian sorites the order is reversed. The 
subject of one premise becomes the predicate of the next, 
and the conclusion has for its subject the subject of the 
last premise and for its predicate the predicate of the first 
premise. 

In the Aristotelian sorites we go from the term of least 
extent to the term of greatest extent, and in the Goclenian 
sorites from the term of greatest extent to the term of 
least extent. Therefore the former is also called an as- 
cending sorites and the latter a descending sorites. 



SYLLOGISMS. 61 



EXAMPLES. 

Aristotelian sorites. 
All flies are insects. 
All insects are invertebrates. 
All invertebrates are animals. 
All animals are organic beings. 
.*. All flies are organic beings. 

Goclenian sorites. 
All animals are organic beings. 
All invertebrates are animals. 
All insects are invertebrates. 
All flies are insects. 
.». All flies are organic beings. 

In regard to the quality and quantity of the premises, 
it should be observed that in the Aristotelian sorites the 
only premise that may be particular is the first, and the 
only one that may be negative is the last. The Aristo- 
telian sorites given above may be put into the syllogistic 
form as follows : 

1. 2. 3. 

B is C C is D D is E 

A is B A is C A is D 

.•. A is C .-. A is D .-. A is E 

The simple syllogisms of which the sorites is composed 
are all in the first figure, and in this figure the major 
premise must be universal and the minor premise affirma- 



62 OUTLINES OF LOGIC. 

five. Hence the first premise of the sorites, being the 
only minor premise expressed, is the only one that may 
be particular. 

Again, the last premise of the sorites is the only one 
that may be negative. For if any other be negative, the 
conclusion of the corresponding simple syllogism would 
be negative, and as this conclusion is to be used as the 
minor premise of the next syllogism, we would have a 
syllogism in the first figure having a negative minor 
premise, which is contrary to the rule. 



CHAPTER V. 



FALLACIES. 



1. Definition of Fallacy. 



A fallacy is an argument which at first sight appears 
to he valid, hut in reality violates the rules of the syllogism,. 

2. Classification of Fallacies. 

Fallacies are usually divided into two classes : logical 
fallacies and material fallacies. 

a) A logical fallacy is one in which the premises are 
insufficient or where the conclusion does not follow from 
the premises. 

h) A material fallacy is one in which the premises are 
sufficient for the conclusion, hut in which either the truth 
of the premises remains to he proved or the conclusion is 
irrelevant to the point that is to he demonstrated. 

A material fallacy is not a fallacy in the form, but in 
the suhject-matfer. To decide whether the premises are 
true or not, is something that logic cannot do. The sub- 
ject of material fallacies is therefore one with which logic 
is only indirectly concerned. 

(63) 



64 OUTLINES OF LOGIC. 

3. Logical Fallacies. 

Logical fallacies may be divided into purely logical and 
semi-logical. 

Of purely logical fallacies, including all the distinct vio- 
lations of the syllogism, the following may be mentioned : 

1. Fallacy of Four Terms. 

2. Fallacy of Undistributed Middle. 

3. Fallacy of Illicit Process of either Major or Minor Term. 

4. Fallacy of Negative Premises. 

5. Fallacy of Particular Premises. 

All these fallacies are explained in the chapter treating 
of syllogisms. We will only give the following examples : 

A is B. 

1. C is D. 

.-. D is A. 

Here we have no middle term or medium of comparison 
between A and D. Hence in order to compare A and D 
two syllogisms are required, one for comparing A and C 
with B and the other for comparing A and C with D. 

All birds are vertebrates. 

2. All fishes are vertebrates. 
.*. All fishes are birds. 

All insects are animals. 

3. No dogs are insects. 
.*. No dogs are animals. 



FALLACIES. 65 



No birds are quadrupeds. 

4. No horses are birds. 

.*. No horses are quadrupeds. 

Some flowers are blue. 

5. Some flowers are red. 

/. Some red things are blue. 

Of semi-logical fallacies the more common are : 

1. Fallacy of Equivocation. 

2. Fallacy of Composition. 

3. Fallacy of Division. 

4. Fallacy of Accident. 

5. Converse Fallacy of Accident. 

6. Fallacy of Many Questions. 

7. Fallacy of Amphibology. 

8. Fallacy of Positive and Negative Intention. 

1. Fallacy of Equivocation. — This fallacy consists in 
using a term in two (liferent senses. 

In most cases it is the middle term that is used in two 
different significations in the premises. In such a case the 
fallacy is usually called a fallacy of ambiguous middle. 
The fallacy of equivocation is, in reality, a fallacy of four 
terms, as is easily seen by substituting some other expres- 

—5 



66 OUTLINES OF LOGIC. 

sion for the ambiguous term in each premise. For in- 
stance, 

No designing person ought to be trusted. 

Engravers are designers. 
.*. Engravers ought not to be trusted. 

A ball is a round body. 
He attended the ball. 
.*. He attended a round body. 

2. Fallacy of Composition. — This fallacy consists in 
using the middle term, distributively in the major 'premise 
and collectively in the minor premise. 

For instance, 

Five and three are two numbers. 
Eight is five and three. 
.•. Eight is two numbers. 

3. Fallacy of Division.— This fallacy consists in using 
the middle term collectively in the major premise and dis- 
tributively in the minor premise. 

For instance, 

Eight is one number. 
Five and three are eight. 
.*. Five and three are one number. 

All the apples in the garden are worth one hun- 
dred dollars. 
This is one of the apples in the garden. 
.*. This apple is worth one hundred dollars. 



FALLACIES. (tf 



4. Fallacy of Accident. — This fallacy consists in as- 
sorting of something described by some accidental pecul- 
iarity what is true only of its substance. 

For instance, 

What yon bought yesterday you eat to-day. 
You bought raw meat yesterday. 
.*. You eat raw meat to-day. 

We do not buy meat because it is raw, but because it is 
meat. That the meat is raw is only an accidental prop- 
erty. 

5. Converse Fallacy of Accident. — This fallacy con- 
sists in arguing from a special case to a general one. 

For instance, 

. Alcohol acts as a poison when used in excess. 
.*. Alcohol is always a poison. 

G. Fallacy of Many Questions. — This fallacy consists 
in combining two or more questions into one to which a 
single answer cannot be given. 

Thus, if a man who has never used tobacco is asked If 
he has given up smoking, he can neither answer the ques- 
tion affirmatively nor negatively. This question would 
namely imply that he did smoke. This fallacy arises from 
the fact that though only one question is expressed, two or 
more questions are implied. 



68 OUTLINES OF LOGIC. 



7. Fallacy of Amphibology. — This fallacy consists in 
ambiguity in the grammatical structure of a sentence, by 
tohich it r rnay have two or more different meanings. 

Thus a word may be used so as to leave it ambiguous 
whether it is subject or predicate, or the reference of a 
pronoun or an adverb may be ambiguous. For instance, 

He likes me better than you. 

We also get salt from the ocean, which is very 
useful to man. 

He promised his father to help his friends. 

8. Fallacy of Positive and Negative Intention. — 
This fallacy consists in using certain negative words, as 
no and nothing, in two different senses. 

For instance, 

No cat has two tails. 
Every cat has one tail more than no cat. 
.*-. Every cat has three tails. 

Nothing is better than happiness. 
Bread is better than nothing. 
,\ Bread is better than happiness. 

4. Material Fallacies. 
Of material fallacies the more common are: 
1. Begging the Question (Petitio principii). 



FALLACIES. 69 



2. Fallacy of False Cause (Non causa pro causa). 

3. Fallacy of Irrelevant Conclusion (Ignoratio elenclii). 

1. Begging the Question. — Tl\is fallacy consists in 
using as a premise either the conclusion itself or some con- 
sequence of the conclusion which is to be established. 

Another name for this kind of fallacy is arguing in a 
circle (circulus in demonstrando). Thus we argue in a 
circle if we try to prove the existence of God in the fol- 
lowing way : 

The Scriptures must be true, as they are the word 
of God. 

The Scriptures declare that God exists. 

.*. God exists. 

Here we prove that God exists from the truth of the 
Scriptures and prove the truth of the Scriptures from the 
fact that they are the word of God, which evidently im- 
plies that we take for granted what is to be proved, namely, 
that God exists. 

2. Fallacy of False Cause. — This fallacy consists in, 
assigning as a cause something that, in reality, has noth- 
ing to do with the conclusion. 

If one event occurs shortly before another event or 
they occur at the same time, and if we take the mere con- 
junction of the two events as a satisfactory proof that one 
is the cause of the other, we commit a fallacy of false 



70 OUTLINES OF LOGIC. 

cause. Two events may be simultaneous without having 
the least relation. 

3. Fallacy of Irrelevant Conclusion. — This fallacy 
consists in arriving at a conclusion different from the one 
that is to he established. 

Suppose we had to prove that all the angles of a tri- 
angle are together equal to two right angles, and we only 
proved that they cannot be less than two right angles. 
That would be a fallacy of irrelevant conclusion, for the 
proposition would not be proved before we had also proved 
that the angles cannot be more than two right angles. 

The fallacy of irrelevant conclusion is one of the most 
common of the material fallacies, and is known under 
various names. Of the more common forms of this kind 
of fallacy the following two may be mentioned : 

a) Argumentum ad hominem, which consists in making 
an appeal to the vanity or prejudice of our opponent so 
as to make him blind to the unreasonableness of the argu- 
ment. 

b) Argumentum ad poptdum, which differs from the 
former fallacy only in being addressed to a body of people 
instead of one individual. 

5. Paralogisms and Sophisms. 

Fallacies may also be divided into paralogisms and 
sophisms. 



FALL AC FES. 71 



1. A paralogism is an undesigned fallacy, the person 
that commits it being unconscious of the falsity of his 
argument. 

2. A sophism is a fallacy which is consciously used to 
deceive. 



CHAPTER VI. 



METHOD. 



1. Science. 



1. Definition of Science. — -Science is classified knowl- 
edge. 

A person may have learned a good many facts about a 
certain group of objects or phenomena, but in order that 
his knowledge may be entitled to the name of scientific 
knowledge, the facts must be arranged according to cer- 
tain principles and the relation between them clearly un- 
derstood. 

Scientific knowledge does not differ in kind from com- 
mon knowledge, as the powers used in acquiring knowl- 
edge, whether it be common or scientific, must obviously 
be the same. They differ only in degree of accuracy. 

2. Requisites of a Science. — The requisites of a science 
are: 

a) All statements made must be true. 

b) A science should be as general as possible ; i. e., the 
process of generalization should be carried as far as possi- 
ble. 

(72) 



METHOD. 73 



c) In every science there should he a certain order, and a 
necessary connection between the various elements of the 
science. 

d) The number of facts ascertained should be as great 
as possible. 

3. Axioms. — An axiom is a self-evident and intuitively 
true proposition. 

The truth of an axiom cannot and need not be demon- 
strated by any simpler propositions. 

The ultimate principles of all deductive sciences are 
axioms, which form the basis on which all the demonstra- 
tions of those sciences are founded. As examples of axi- 
oms we may mention the following two : 

The whole is greater than its parts. 

Things that are equal to the same thing are equal 
to each other. 

2. Deduction and Induction. 

1. Definition of Method. — Method is a certain mode 
of procedure for arriving at a certain result. 

Method must be used in all sciences, though th.e kind of 
method which is to be used will be different for different 
sciences. 

The methods used in science may be classified under 
the two heads, deduction and induction. 



74 OUTLINES OF LOGIC. 

2. Deduction. — Deduction is the process of deriving a 
particular truth from a general truth. 

In the deductive method we proceed from the general 
to the particulars which are embraced in it. 

For instance, 

All insects are animals. 
All butterflies are insects. 
.-. All butterflies are animals. 

Here we first state a general truth, something that is 
true about all insects, namely, that they are arfimals. Then 
we proceed to analyze this general truth into the partic- 
ulars it embraces, and finally we reach a conclusion con- 
cerning one of the particulars, namely, butterflies. 

The deductive method is also called the analytic method. 

3. Induction. — Induction is the process of deriving gen- 
eral truths from particular ti^uths. 

In the inductive method we proceed from the observa- 
tion of particular truths or facts to the establishment of 
general laws. As an example of inductive reasoning we 
give the following : 

By observations we know that Mercury, Venus, the 
Earth, Mars, Jupiter, Saturn, Uranus and Nep- 
tune move around the sun in elliptic orbits. 

Hence all the planets move around the sun in 
elliptic orbits. 



METHOD. 75 

The inductive method is also called the syntlietical 
method. 

Induction is of two kinds: Perfect induction and im- 
perfect induction,. 

a) The induction is perfect when all the particular cases 
have been examined. 

For instance, 

Mercury, Venus, the Earth, Mars, etc., move in el- 
liptic orbits around the sun. 

Hence all the known planets move around the sun 
in elliptic orbits. 

In the conclusion we affirm something only of the par- 
ticular cases that have been examined. We do not say 
that all planets move in elliptic orbits around the sun, but 
only all the known planets. The conclusion must there- 
fore be certain. 

Perfect induction always leads to a necessary and cer- 
tain conclusion. 

b) The induction is imperfect when we have examined 
only some of the particular cases and from them infer a 
general law. 

In the first example given above we assert of all planets 
something that has been found to be true of all the known 
planets. Hence we infer that if some new planet would 
be discovered it would most likely move in an elliptic 



76 OUTLINES OF LOGIC. 

orbit around the sun like the planets that are now known. 
This conclusion is very probable, but not certain. 

Imperfect induction can never lead to a certain and 
necessary conclusion, but only to a probable conclusion. 

3. Definition. 

1. Definition defined. — To define a thing is to give 
those attributes by which it differs from all other things, 
and the process is called logical definition. 

To define something means to state what it is, or to 
distinguish it from all other things. It is not necessary, 
however, to enumerate all the attributes belonging to the 
thing which is to be defined, but only the essential attri- 
butes. The essential attributes are the genus and the dif- 
ferentia. 

a) By the genus is meant the next higher genus of which 
the thing to be defined is a species. 

b) By the differentia is meant those specific characters 
by which the thing to be defined differs from all other spe- 
cies of the same genus. 

Definition thus consists in giving the genus and the 
differentia of the thing to be defined. A definition has 
the form of a categorical judgment, of which the subject 
is the thing to be defined, and the predicate the genus and 
the differentia. 

Suppose we want to define an equilateral triangle. An 
equilateral triangle is a species of the genus triangle, and 



METHOD. 77 



differs from all other triangles in having the three sides 
equal. Hence the genus is triangle and the differentia 
having the three sides equal. The definition of an equilat- 
eral triangle will then be : 

An equilateral triangle is a triangle 

genus. 
having its sides equal. 

differentia. 
It should be observed that it is essential for a logical 
definition that the genus should be the next higher genus. 
Hence the following definition is not correct: 

An equilateral triangle is a plane figure having its 
sides equal. 
Plane figure is not the next higher genus. 
We will give two more examples of definitions, viz. : 
Man is a rational animal. 

differentia genua 
A parallelogram is a quadrilateral 

genus 

whose opposite sides are parallel. 



differentia. 

2. Rules for Definition. — In definition the following 
rules should be observed : 

I. The definition should he adequate, i. e., neither too 
wide nor too narrow. 

a) The definition is too wide if the predicate has greater 



78 OUTLINES OF LOGIC. 

extent than the subject, i. e., if it includes other things be- 
sides those that are to be defined. 

For instance, 

A bird is an animal that has a backbone. 

This definition is too wide because also fishes, reptiles 
and mammals have a backbone. 

Man is a rational being. 

This is also too wide, because rational being also in- 
cludes God. 

h) The definition is too narrow if the subject has a 
greater extent than the predicate, i. e., if it excludes some 
of the things that are to be defined. 

For instance, 

A triangle is a figure having three equal sides. 

This definition is too narrow, because all isosceles and 
scalene triangles are excluded. 

A bird is a feathered animal that sings. 

This is also too narrow. Some birds do not sing. 

The test of an adequate definition is that it may be both 
simply converted and converted by contraposition. If the 
definition is too wide, it cannot be simply converted. If 
it is too narrow, it cannot be converted by contraposition. 

II. The definition should not contain the term which is 
to he defined. 

The violation of this rule is called defining in a circle. 
We thus define in a circle if we define law as a lawful 



METHOD. 79 

command, because we use in the definition the word we 
want to define. As another example let us take 
Life is the sum of the vital functions. 
Here we use the term vital, which is really a synonym 
of the term to be defined, and which only can be explained 
by the term life. 

III. The definition should be affirmative. 

The definition should state what a thing is, and not 
what it is not. Hence the following definitions are unsat- 
isfactory : 

A straight line is a line no portion of which is 

curved. 
A regular polygon is one that is not irregular. 
Light is the absence of darkness. 

IV. In definition we should not give any superfluous or 
accidental attributes. 

For instance, 

A pentagon is a polygon having five sides and five 
angles. 
This definition is incorrect, as the latter attribute is 
superfluous. 

A parallelogram is a quadrilateral having the op- 
posite sides parallel and having the opposite 
sides and angles equal. 
Here two attributes are given that follow from the par- 
allelism of the sides, and which therefore are superfluous. 



OUTLINES OF LOGIC. 



A horse is a four-legged animal with a tail and a 
mane. 

Here accidental attributes are used. 

3. Nominal and Heal Definitions. — Definitions are 
divided into nominal and real. 

a) A nominal definition is one v)hich explains the mean- 
ing of the term which is used as the name of the thing. 
For instance, 

A phonograph is an instrument for registering and 

reproducing sound. 
A telephone is an instrument for conveying sound 
to a great distance. 

h) A real definition is one which defines the thing itself. 

Thus a real definition of phonograph would be a treatise 
on the construction and use of that instrument. 

In all scientific investigations it is the aim to obtain 
real definitions, but for many practical purposes nominal 
definitions will be sufficient. 

4. Description. — By description is meant an enumera- 
tion of all the properties of a thing. 

A description of an elephant, for instance, would thus 
consist in the enumeration of all the properties belonging 
to elephants. In definition we give only the essential at- 
tributes of a thing. In description, again, we may use 
not only essential, but also accidental attributes. The 



METHOD. 81 



natural-history sciences furnish good examples of descrip- 
tions. 

4. Division. 

1. Division defined. — By logical division is meant the 

process of dividing a genus into its species according to a 

certain principle of division. 

For instance, 

[ right-angled 

Triangles may be divided into X acute-angled 

' obtuse-angled. 
Here the genus triangle is separated into its three spe- 
cies, and the basis or principle of division, commonly called 
the fun damentum divisionis, is the size of the angles. 

C triangles 
quadrilaterals 
Polygons may be divided into X pentagons 

hexagons 
etc. 
Here the principle of division is the number of sides. 

2. Dichotomy. — If a genus is divided into two species 
each of which is the contradictory of the other, the divi- 
sion is commonly called dichotomy. 

For instance, 

vertebrates 



Animals may be divided into 

( not-vertebrates. 

n i i ■,. . ■, i . ( triangles 

Polygons may be divided into < 

( not-triangles. 



—6 



82 



OUTLINES OF LOGIC. 



Books are divided into J 



Although, from a logical point of view, dichotomy is a 
perfect division, it is for most practical purposes not very 
convenient. 

3. Rules foe Division. — In logical division the follow- 
ing rules should be observed : 

I. In division there should he only one principle of di- 
vision. 

Hence the following divisions are not correct: 

r English 
French 
German 
Quarto 
Octavo 
etc. 

The first division is according to language and the sec- 
ond according to size. 

f isosceles 
equilateral 
right-angled 
acute-angled. 

The first division is according to the relative length of 
the sides and the second according to the size of the an- 
gles. 

Such a division is generally called a cross-division. 

II. The principle of division should he an actual attri- 
bute of the genus which is to he divided. 



Triangles are divided into J 



METHOD. 83 

III. In division the members should exclude each other, 
and they should all be co-ordinate or of the same rank. 

For instance, 

triangles 

quadrilaterals 

parallelograms 

polygons having more 

than four sides. 



Polygons may be divided into -< 



Parallelograms are included in quadrilaterals, and con- 
sequently the members do not exclude each other. 

IV. The division should be complete, i. <?., the sum of 
the species should be equal to the genus. 

Hence no species must be left out. For instance, 

( mammals 
Vertebrates are divided into < birds 

' fishes. 
Here reptiles and batrachians are left out. 

acute-angled 



Triangles are divided into 

right-angled. 

Here obtuse-angled triangles are left out. 



V. In division v)e should proceed from proximate gen- 
era to proximate species. 

We should not proceed from a high genus to a low spe- 
cies, but from the genus to the next lower species. 



84 



OUTLINES OF LOGIC. 



In the following division this rule is violated : 

r horses 
dogs 
Vertebrates are divided into -{ eagles 

lions 
etc. 

A logical division of vertebrates would be into 

mammals 
birds 
fishes 

batrachians 
reptiles. 

Each of these species may further be divided and sub- 
divided until we reach the lowest species. 

4. Partition. — By partition is meant the separation in 
thought of the physical parts of which an individual ob- 
ject is composed. 

For instance, 

Water is composed of oxygen and hydrogen. 

A plant may be divided into root, stem, leaves, etc. 

This mode of separating an object into its constituent 
parts is something with which logic is not directly con- 
cerned, and should not be confounded with logical divi- 
sion. 



METHOD. 85 

5. Demonstration. 

1. Demonstration defined. — Demonstration is an act 
of reasoning by which the truth of a proposition is estab- 
lished as a consequence of other truths. 

In every demonstration we notice: 

a) The proposition that is to be proved. 

b) The premises or grounds of proof. 

c) The necessary connection between the different parts 
of the demonstration. 

The premises are either definitions, axioms, or previously 
established propositions. 

2. Rules for Demonstration. — For demonstration we 
have the following rules : 

I. No proposition, must be used as a premise which is 
not known to be true. 

II. The proposition which is to be proved must not be 
used as a premise. 

III. No proposition whose truth depends on the truth 
of the proposition which is to be proved m ust be used as a 
prem ise. 

IV. There must be no leaps in the demonstration. 

V. We must not prove another proposition instead of 
the one that is to be established. 

For violations of the rules given above see Chapter V 
(Fallacies). 



86 OUTLINES OF LOGIC. 

3. Classification of Demonstrations. — Demonstra- 
tions are divided — 

I. Into direct and indirect. 
II. Into deductive and inductive. 
III. Into a priori and a posteriori. 

I. a) A direct demonstration is one in which the truth 
of a proposition is immediately deduced from certain other 
truths that have already bee?i established. 

b) An indirect demonstration is one in which the truth 
of a proposition is established by proving the absurdity of 
its contradictory. 

In a direct demonstration we give the reasons why the 
conclusion must be true. In an indirect demonstration 
we give the reasons why it cannot be false. In an in- 
direct demonstration we proceed in the following manner. 
We make a supposition contrary to the conclusion which 
is to be proved. From this supposition we deduce a series 
of conclusions until we arrive at a conclusion which is con- 
trary to some known truth. Then by modus tollens we 
conclude from the falsity of the consequent to the falsity 
of the antecedent ; that is, we conclude that the supposi- 
tion made must be false, as it leads to an absurd con- 
clusion. And since this supposition is false, its contra- 
dictory, or the conclusion which is to be established, must 
be true ; because of two contradictories one must be true 
and the other false. 



METHOD. 



87 



We give the following example of an indirect demon- 
stration : 

Two straight lines perpendicular to the same straight 
line are parallel. 



t 




Let the two straight lines AB and CD be both per- 
pendicular to AC; then AB is parallel to CD. 

For suppose that AB is not parallel to CD. Then the 
two lines AB and CD must meet at some point if they be 
produced. Let them meet at the point E. Then there 
will be two perpendiculars, EA and EC, let fall from the 
same point on the same straight line, which is absurd. 
Therefore the two lines, AB and CD, cannot meet if 
they be produced ever so far. Hence the two lines are 
parallel. 

II. a) A deductive demonstration is one in which we 
proceed from the whole to the parts. 



OUTLINES OF LOGIC. 



b) An inductive demonstration is one in vdiich we pro- 
ceed from the parts to the whole. 

In a deductive demonstration we prove that something 
holds true of the whole, and then conclude that it must 
hold true of every part or individual case of the whole. 
In an inductive demonstration, again, we prove that some- 
thing holds true of all the parts or individual cases and 
then conclude that it must hold true of the whole. 

We will give the following example of an inductive 
demonstration : 

An angle inscribed in a segment is measured by half the 
arc included between its sides. 

This proposition admits of three cases: 
1st. Let the centre of the circle be on one of the sides 
of the angle. 

Draw the radius OC. Because 
OC is equal to OB, the angle OBC 
is equal to the angle OCB ; there- 
fore the angles OBC and OCB are 
together double the angle OBC. 
The angle AOC is equal to the sum 
of the angles OBC and OCB. Hence the angle AOC is 
double the angle OBC. But the angle AOC is measured 
by the arc AC. Hence the angle ABC is measured by 
half the arc AC. 




METHOD. 



89 




2d. Let the centre of the circle be within the angle. 

Draw the diameter BD. By the 
first case we know that the angle 
ABD is measured by half the arc AD 
and the angle DBC by half the arc 
DC. Therefore the angle ABC is 
measured by half the sum of the arcs 
AD and DC, i. <?., half the arc AC. 

3d. Let the centre be w.thout the angle. 

Draw the diameter BD. By the 
first case we know that the angle 
ABD is measured by half the arc 
AD, and the angle CBD by half the 
arc CD. Therefore the angle ABC 

ry is measured by half the difference of 

the arcs AD and CD, i. e., half the arc AC. 

Hence the proposition is true for all possible cases, and 
therefore it must be true for any angle inscribed in a seg- 
ment. 

III. a) A demonstration a priori is one in which the 
premises are given by intuition. 

b) A demonstration a posteriori is one in which the 
premises are given by experience. 

In mathematics, for instance, all the relations between 
quantities are established by a chain of reasoning which 
ultimately depends on certain a priori or intuitive priuci- 




90 OUTLINES OF LOGIC. 

pies, namely the ideas of space and' number. In the nat- 
ural sciences, again, the arguments are mainly a posteriori, 
as the premises are given by experience. 

6. Analogy. 

Reasoning by analogy is a process by which we infer 
that if two or more objects are similar in certam respects, 
they will also be similar in other respects. 

Reasoning by analogy gives only a probable conclusion. 
The degree of probability depends on the number of ob- 
served resemblances and the importance of the points in 
which the objects agree. Hence in order that reasoning 
by analogy should be of any value, the attributes that are 
similar should be as many as possible and should not be 
accidental. If it can be shown that one or more of the 
essential attributes of the first object is incompatible with 
some essential attribute of the second object, the argu- 
ment is invalid. 

For instance, 

By observing the similarity between lightning and 
electricity in many respects, Franklin was, by 
analogy, led to the conclusion that they were 
identical. 

The earth and the planet Mars resemble each other 

in many respects. 
Hence Mars is probably inhabited. 



METHOD. 91 



7. Hypothesis. 

A hypothesis is a supposition made to account for a cer- 
tain group of phenomena. 

The probability of a hypothesis depends on the number 
of facts or phenomena that may be explained by it. The 
greater the number of phenomena it will explain, the more 
we are justified to believe the hypothesis to be right. 

As examples of hypotheses we may mention Laplace's 
Nebular Hypothesis to explain the formation of the solar 
system, and the Copernican theory of the solar system. 

8. Classification of Sciences. 

From a formal point of view the sciences are usually 
divided into empirical and rational . 

The difference between the empirical and the rational 
sciences is given in the following schedule : 
Data : facts. 
Aim: the establishment of general 

laws. 
Method: mainly inductive. 
Data: universal principles. 



Empirical 



Aim: the establishment of particu- 
Rational -I 

lar truths. 

I Method: mainly deductive. 

Botany, zoology, chemistry and geology are examples 

of empirical sciences. Mathematics is an example of a 

rational science. 



EXEEOISES. 



CHAPTER II. 

CLASSIFICATION OF CONCEPTS. 

1. For each one of the following concepts state whether 
it is positive or negative, absolute or relative, concrete or 
abstract : 

Book Man Daughter 

Father House Metal 

Weight Darkness Independence 

Holiness Logic Whiteness 

Unnatural Light Son 

Air Resemblance Animal 

Oblique-angled Curved Straight 

Being Reason Rational 

Figure Triangle God 

EXTENT AND CONTENT. 

1. In each one of the following pairs of concepts state 
which concept has the greater extent, and which has the 
greater content : 

( Dog I Plant t \ Man 

j Animal J ( Tree | Being 

(93) 



EXERCISES. 



93 



10 



j Heavenly body ( Element j Eagle 

4 1 Planet 5 ( Metal 6 (Bird 
| Equitable triangle I Fish I Rock 

\ Equiangular triangle ( Vertebrate \ Granite 
Fly ( Book ( House 



\ Insect 



I Dictionary ( Brick 1 



louse 



2. Arrange the following terms in several series in such 
a manner that the first term of each series shall have the 
greatest extent and the last term the least extent. 



Salmon 


Plant 


Europeans 


Polygon 


Fish 


Square 


Man 


Animal 


Figure 


Apple tree 


Vertebrate 


Rational being 


Plane figure 


Quadrilateral 


Phcenogam 



RELATION OF CONCEPTS. 



1. State the relation between the concepts of each of 



the 

■I 



following groups: 






Plant 
Organic being 


2 J Polygon 
u \ Figure 


\ Flies 
] Bees 


Square 
Parallelogram 


I Man 

I American 


( European 
\ Italian 


Metal 
Not-metal 


I Gold 
( Iron 


( Salmon 
9 ] Fish 


Bird 
Reptile 


\ Straight 
| Not-straight 


12 | Eagle 
| Sparrow 



94 OUTLINES OF LOGIC. 

CHAPTER III. 

CLASSIFICATION OF JUDGMENTS. 

State the logical character as to quality, quantity, rela- 
tion and modality of each of the following judgments: 

All triangles are figures. 
If he is honest, he should speak the truth. 
Triangles are divided into right-angled and oblique- 
angled. 
The table is black. 
If rain has fallen, the ground is wet. 
Napoleon was a great man. 
No triangles are squares. 
Some angles are obtuse. 
Some horses are not black. 
His character is either good or bad. 
Iron is an element. 
Some men are good. 
God is omniscient. 
Some men are not kings. 
This horse is not black. 
Some triangles are equilateral. 
No planets are self-luminous. 
Some of our muscles are involuntary. 
New York is a city. 
Horses are vertebrate animals. 



EXERCISES. 95 



IMMEDIATE INFERENCE. 

1. Which of the four judgments A, I, E and are true 

or false when 

1. A is true 5. E is true 

2. A is false 6. ^ is false 

3. / is true 7. O is true 

4. / is false 8. is false 

2. Convert the following judgments: 

All vertebrates are animals. 

Some poisonous things are plants. 

No men are angels. 

Man is mortal. 

Some persons are wise. 

Some quadrupeds are not horses. 

Some birds are eagles. 

No plants are animals. 

All triangles have three sides. 

No triangles are quadrilaterals. 

3. If the judgment 

Some triangles are not figures 

is false, how could you prove the truth of the judg- 
ment 

Some triangles are figures? 

4. How can you conclude from the falsity of the judg- 

ment 

No animals living in water are fishes 
to the truth of 

Some fishes live in water ? 



96 



OUTLINES OF LOGIC. 



5. How can you conclude from the truth of the judgment 

No insects are vertebrates 
to the falsity of 

Some vertebrates are insects ? 

6. How can you prove the falsity of the judgment 

No not-triangles are figures 
from the truth of 

Some figures are not triangles ? 



CHAPTER IV. 



SIMPLE SYLLOGISMS. 

Construct syllogisms from the terms given in each of 
the following moods : 

1st Figure. 



[ P = animal 
Barbara < M = bird 
(. S = eagle 
[ P= mortal 
Darii-l M = man 
( S = being 



i F = irrational 

Celarent \ M = man 

I S = American 

( P = square 

Ferio\ M = triangle 

\ S = equilateral 

figure 



2d Figure. 
( P = animal ( P=insect 

Cesarel M=plant Ca?nestres< M=animal 

( S=grass I S=rock 



EXERCISES. 



97 



( P=trapezium ( P=fixed star 

Festino ■! M=parallelogram Barofro < M=selflnminous 
[ S=quadrilateral I S=heavenly 

body 

3d Figure. 

j P=mao ( P=hawk 

Darapti\ M= American Disamis-l M= vertebrate 
\ S=rational I S=animal 



( P= mortal 

Datisl < M=man 
( S^black 



( P = herb 

Felapton < M = tree 

[ S = plant 



( P = having feet ( P = triangle 

Bokardo < M = reptile Ferison < M = pentagon 

lS = animal (. S = equilateral 



4th Figure. 

( P = granite 
Bvamantip < M = rock Camenes 

\ S = inorganic 



( P = yellow 
Dimaris\ M = butterfly Fesap 
v S = insect 



P = European 
M = man 
S = plant 

P = cat 

M = dog 
S = animal 



-\ M = 



Fresison 



f P = square 
M = hexagons 
S = equilateral 



03 



OUTLINES OF LOGIC. 



COMPOUND 


SYLLOGISMS. 


1. From the following terms 


constru 


ct compound syllo- 


gisms, epichiremata, and sorites 


: 




1 - 


r Organic being 
Plant 

Ph sen ogam 
Tree 
Oak 




^ 


Animal 
Vertebrate 
Reptile 
Crocodile 


,. 


Figure 
Plane figure 
Polygon 
Triangle 
^ Isosceles triangle 




4 - 


Being 
Man 

European 
German 



2. Construct enthymes by taking any three consecutive 
terms of those given above. 



CHAPTER V. 



FALLACIES. 

Point out the fallacies in the following arguments 

1. Some plants are trees. 
Some plants are grasses. 
,\ Some grasses are trees. 



EXERCISES. 99 



2. Red is a color. 
Bine is a color. 

,\ Bine is red. 

3. All men are rational beings. 
All men are animals. 

,\ All animals are rational beings. 

4. All men are organic beings. 
No dogs are men. 

,\ No dogs are organic beings. 

5. All moral beings are accountable. 
No brute is a moral being. 

,\ No brute is accountable. 

6. Design implies a designer. 

The universe abounds in design. 
.'. God exists. 

7. A stone is a body. 
An animal is a body. 
Man is an animal. 

,\ Man is a stone. 

8. Nothing is better than wisdom. 
A dime is better than nothing. 

,', A dime is better than wisdom. 



100 OUTLINES OF LOGIC. 



9. Metals are elements. 

Iron is a metal. 

/. Iron is an element. 

10. If this medicine is of any value, those who take 

it will improve in health. 
I have taken it, and have improved in health. 
,\ This medicine is of value. 

11. Dickens's Oliver Twist is one of the books in 

the book-store of my friend. 
I have bought Dickens's Oliver Twist. 
,\ I have bought one of the books in my friend's 
book-store. 

12. His books are worth one hundred dollars. 

Shakespeare is one of his books. 
.•. Shakespeare is worth one hundred dollars. 

13. The people of the city are suffering from the 

yellow fever. 
You are one of the people of the city. 
,\ You are suffering from the yellow fever. 

14. Light is contrary to darkness. 
Feathers are light. 

.*. Feathers are contrary to darkness. 



EXERCISES. 



101 



CHAPTER VI. 



DEFINITION. 



1. Define the following terms, and point out the genus 
and the differentia in each definition : 



Element 


Capita] 


Dictionary 


Nonagon 


Vertebrate 


Genus 


Logic 


Animal 


Species 


Science 


Man 


Hypothesis 


Syllogism 


Rhombus 


Circle 


Deduction 


Plant 


Straight line 


Induction 


Parallelogran 


i Judgment 



2. What rules do the following definitions violate? 

1. A straight line is one no portion of which is 

curved. 

2. A rectangle is a figure having four right angles. 

3. A trapezium is a quadrilateral having the oppo- 

site sides parallel. 

4. A hexagon is a figure having six equal sides. 

5. A mammal is an animal that does not reproduce 

its species by laying eggs. 

6. A square is a four-sided figure with equal sides. 

7. Evil is that which is not good. 



102 



OUTLINES OF LOGIC. 



DIVISION. 

In what are the following divisions faulty '( 



Cryptogams 
Monopetalons 
Apetalous 
Polypetalous 

Men 



1. Plants are divided into -< 



2. Mankind may be divided into J Women 

I Children 

f Sea-birds 
Sparrows 

3. Birds are divided into -<{ Eagles 

Parrots 
Gallinaceous birds 

4. The faculties of the f Perce P tio « 



mind are divided into 



5. Books are divided into -< 



Imagination 
I Keason 

Grammars 

Dictionaries 

French 

German 

Italian 



